This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A378806 #5 Dec 08 2024 02:44:28
%S A378806 2,9,0,8,8,2,0,7,1,5,2,1,2,8,7,2,1,2,7,6,2,5,9,7,2,5,6,6,8,6,8,1,0,3,
%T A378806 5,7,7,3,3,6,8,1,7,6,1,6,7,6,0,9,7,9,2,7,5,8,2,3,7,9,3,5,9,2,6,2,2,8,
%U A378806 4,8,1,2,4,6,8,0,2,5,4,2,5,5,0,5,5,9,3,3,9,1,8,9,7,1,6,4,9,5,6,0,3,0,3,3,4
%N A378806 Decimal expansion of Sum_{k>=1} 1/binomial(4*k, k).
%H A378806 Necdet Batir and Anthony Sofo, <a href="http://dx.doi.org/10.1016/j.amc.2013.05.053">On some series involving reciprocals of binomial coefficients</a>, Appl. Math. Comp., Vol. 220 (2013), pp. 331-338.
%F A378806 Equals 4F3(1, 4/3, 5/3, 2; 5/4, 3/2, 7/4; 27/256) / 4, where 4F3 is a generalized hypergeometric function.
%F A378806 Equals 27*c^2/((c^2-4)*(2*c^2+1)^2) + (3*c*(c^2-1)*(2*c^2-1)/(2*(2*c^2+1)^3)) * log((c-1)/(c+1)) + (3*(c^2-1)*(2*c^4-2*c^3-7*c^2-3*c+1)/(4*c*(2*c^2+1)^3)) * (c/(c+2))^(3/2) * arctan(2*sqrt(c^2+2*c)/(c^2+2*c-1)) + (3*(c^2-1)*(2*c^4+2*c^3-7*c^2+3*c+1)/(4*c*(2*c^2+1)^3)) * (c/(c-2))^(3/2) * arctan(2*sqrt(c^2-2*c)/(c^2-2*c-1)), where c = sqrt(1 + (16/sqrt(3))*cos(arctan(sqrt(229/27))/3)) (Batir and Sofo, 2013, p. 337, Example 9).
%e A378806 0.29088207152128721276259725668681035773368176167609...
%t A378806 RealDigits[HypergeometricPFQ[{1, 4/3, 5/3, 2}, {5/4, 3/2, 7/4}, 27/256] / 4, 10, 120][[1]]
%Y A378806 Cf. A005810, A073016, A229705, A378807.
%K A378806 nonn,cons
%O A378806 0,1
%A A378806 _Amiram Eldar_, Dec 07 2024